Henrik Kalisch

Singular solutions of a fully nonlinear 2 × 2 system of conservation laws

Existence and admissibility of δ-shock solutions is discussed for the non-convex strictly hyper- bolic system of equations ∂tu + ∂x( 1 (u 2 + v 2 )) = 0, ∂tv + ∂x(v(u − 1)) = 0. The system is fully nonlinear, i.e. it is nonlinear with respect to both unknowns, and it does not admit the classical Lax-admissible solution for certain Riemann problems. By introducing complex-valued cor- rections in the framework of the weak asymptotic method, we show that a compressive δ-shock solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued, and the solutions can be seen to fit into the framework of weak singular solutions defined by Danilov and Shelkovich. Indeed, in this context, we can show that every 2 × 2 system of conservation laws admits δ-shock solutions.

The Whitham Equation as a model for surface water waves

The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the KdV equation. In this work, we identify a scaling regime in which the Whitham equation can be derived from the Hamiltonian theory of surface water waves. The Whitham equation is integrated numerically, and it is shown that the equation gives a close approximation of inviscid free surface dynamics as described by the Euler equations. The performance of the Whitham equation as a model for free surface dynamics is also compared to two standard free surface models: the KdV and the BBM equation. It is found that in a wide parameter range of amplitudes and wavelengths, the Whitham equation performs on par with or better than both the KdV and BBM equations.

Mechanical Balance Laws for Boussinesq Models of Surface Water Waves

Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and it appears that the expressions for many of these quantities are not known. This work presents a systematic derivation of mass, momentum and energy densities and fluxes associated with a general family of Boussinesq systems. The derivation is based on a reconstruction of the velocity field and the pressure in the fluid column below the free surface, and the derivation of differential balance equations which are of the same asymptotic validity as the evolution equations. It is shown that all these mechanical quantities can be expressed in terms of the principal dependent variables of the Boussinesq system: the surface excursion η and the horizontal velocity w at a given level in the fluid.

Convective mixing influenced by the capillary transition zone

Convective mixing of dissolved carbon dioxide (CO2) with formation brine has been shown to be a significant factor for the rate of dissolution of CO2 and thus for determining the viability of geological CO2 storage sites. In most previous convection investigations, a no-flow boundary condition was used to represent the interface between an upper region with CO2 and brine and the single-phase brine region beneath. However, due to interfacial tension between the phases, the water phase is partly mobile in the upper region and advection may occur. Based on linear stability analysis and numerical simulations, we show that advection across the interface leads to considerable destabilization of the system. In particular, the time of onset of instability is reduced by a factor of two and the rate of dissolution is enhanced by a factor of two for three of four formations we consider, and by 40 % for the fourth formation. It is found that exponential decay of the relative permeability away from the interface provides a useful approximation to the real system. In addition, the exponential decay also simplifies the linear stability analysis. Interestingly, formations with large absolute permeability and small porosity have the largest impact from the transition zone, despite the fact that the relative permeability decays quickly above the interface in these formations. This is because the length-scale of instability is smallest in these formations.

Periodic traveling water waves with isobaric streamlines

Abstract It is shown that in water of finite depth, there are no periodic traveling waves with the property that the pressure in the underlying fluid flow is constant along streamlines. In the case of infinite depth, there is only one such solution, which is due to Gerstner.

Boussinesq modeling of surface waves due to underwater landslides

Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion which govern the evolution of the barycenter of the landslide mass include various dissipative effects due to bottom friction, internal energy dissipation, and viscous drag. The surface waves are studied in the Boussinesq scaling, with time-dependent bathymetry. A numerical model for the Boussinesq equations is introduced which is able to handle time-dependent bottom topography, and the equations of motion for the landslide and surface waves are solved simultaneously. The numerical solver for the Boussinesq equations can also be restricted to implement a shallow-water solver, and the shallow-water and Boussinesq configurations are compared. A particular bathymetry is chosen to illustrate the general method, and it is found that the Boussinesq system predicts larger wave run-up than the shallow-water theory in the example treated in this paper. It is also found that the finite fluid domain has a significant impact on the behavior of the wave run-up.

Derivation and comparison of model equations for interfacial capillary-gravity waves in deep water

A matched asymptotic expansion is used to give a formal derivation of a number of systems of model equations for the evolution of interfacial waves subject to capillarity. For one of these systems, approximate solitary waves are found numerically, and the solutions are compared to the Benjamin equation which arises in the special case of one-way propagation.

A boundary value problem for the KdV equation: Comparison of finite-difference and Chebyshev methods

Solutions of a boundary value problem for the Korteweg-de Vries equation are approximated numerically using a finite-difference method, and a collocation method based on Chebyshev polynomials. The performance of the two methods is compared using exact solutions that are exponentially small at the boundaries. The Chebyshev method is found to be more efficient.

A kinematic conservation law in free surface flow

The Green-Naghdi system is used to model highly nonlinear weakly dispersive waves propagating at the surface of a shallow layer of a perfect fluid. The system has three associated conservation laws which describe the conservation of mass, momentum, and energy due to the surface wave motion. In addition, the system features a fourth conservation law which is the main focus of this note. It is shown how this fourth conservation law can be interpreted in terms of a concrete kinematic quantity connected to the evolution of the tangent velocity at the free surface. The equation for the tangent velocity is first derived for the full Euler equations in both two and three dimensional flows, and in both cases, it gives rise to an approximate balance law in the Green-Naghdi theory which turns out to be identical to the fourth conservation law for this system.

A numerical study of the Whitham equation as a model for steady surface water waves

The object of this article is the comparison of numerical solutions of the so-called Whitham equation to numerical approximations of solutions of the full Euler free-surface water-wave problem. The Whitham equation ? t + 3 2 c 0 h 0 ? ? x + K h 0 ? ? x = 0 was proposed by Whitham (1967) as an alternative to the KdV equation for the description of wave motion at the surface of a perfect fluid by simplified evolution equations, but the accuracy of this equation as a water wave model has not been investigated to date.In order to understand whether the Whitham equation is a viable water wave model, numerical approximations of periodic solutions of the KdV and Whitham equation are compared to numerical solutions of the surface water wave problem given by the full Euler equations with free surface boundary conditions, computed by a novel Spectral Element Method technique. The bifurcation curves for these three models are compared in the phase velocity-waveheight parameter space, and wave profiles are compared for different wavelengths and waveheights. It is found that for small wavelengths, the steady Whitham waves compare more favorably to the Euler waves than the KdV waves. For larger wavelengths, the KdV waves appear to be a better approximation of the Euler waves.